A finite element method for nearly incompressible elasticity problems

Braess, Dietrich; Ming, Pingbing

doi:10.1090/S0025-5718-04-01662-X

A finite element method for nearly incompressible elasticity problems
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by Dietrich Braess and Pingbing Ming PDF

Math. Comp. 74 (2005), 25-52 Request permission

Abstract:

A finite element method is considered for dealing with nearly incompressible material. In the case of large deformations the nonlinear character of the volumetric contribution has to be taken into account. The proposed mixed method avoids volumetric locking also in this case and is robust for $\lambda \to \infty$ (with $\lambda$ being the well-known Lamé constant). Error estimates for the $L^{\infty }$-norm are crucial in the control of the nonlinear terms.

References

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI 10.1002/cpa.3160170104
Ivo Babuška and Manil Suri, On locking and robustness in the finite element method, SIAM J. Numer. Anal. 29 (1992), no. 5, 1261–1293. MR 1182731, DOI 10.1137/0729075
C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35 (1998), no. 5, 1893–1916. MR 1639966, DOI 10.1137/S0036142995293766
J. M. Boland and R. A. Nicolaides, Stable and semistable low order finite elements for viscous flows, SIAM J. Numer. Anal. 22 (1985), no. 3, 474–492. MR 787571, DOI 10.1137/0722028
Dietrich Braess, Stability of saddle point problems with penalty, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 6, 731–742 (English, with English and French summaries). MR 1419936, DOI 10.1051/m2an/1996300607311
Dietrich Braess, Enhanced assumed strain elements and locking in membrane problems, Comput. Methods Appl. Mech. Engrg. 165 (1998), no. 1-4, 155–174. MR 1663532, DOI 10.1016/S0045-7825(98)00037-1
Dietrich Braess, Finite elements, 2nd ed., Cambridge University Press, Cambridge, 2001. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German edition by Larry L. Schumaker. MR 1827293
D. Braess, C. Carstensen, and B. D. Reddy, Uniform convergence and a posteriori error estimators for the enhanced strain finite element method, Numer. Math. 96 (2004) 461–479.
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
Manfred Dobrowolski, A mixed finite element method for approximating incompressible materials, SIAM J. Numer. Anal. 29 (1992), no. 2, 365–389. MR 1154271, DOI 10.1137/0729023
R. Durán, R. H. Nochetto, and Jun Ping Wang, Sharp maximum norm error estimates for finite element approximations of the Stokes problem in $2$-D, Math. Comp. 51 (1988), no. 184, 491–506. MR 935076, DOI 10.1090/S0025-5718-1988-0935076-9
Michel Fortin, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér. 11 (1977), no. 4, 341–354, iii (English, with French summary). MR 464543, DOI 10.1051/m2an/1977110403411
J. Frehse and R. Rannacher, Eine $L^{1}$-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975) Bonn. Math. Schrift., No. 89, Inst. Angew. Math., Univ. Bonn, Bonn, 1976, pp. 92–114 (German, with English summary). MR 0471370
Jens Frehse and Rolf Rannacher, Asymptotic $L^{\infty }$-error estimates for linear finite element approximations of quasilinear boundary value problems, SIAM J. Numer. Anal. 15 (1978), no. 2, 418–431. MR 502037, DOI 10.1137/0715026
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
Hervé Le Dret, Constitutive laws and existence questions in incompressible nonlinear elasticity, J. Elasticity 15 (1985), no. 4, 369–387. MR 817376, DOI 10.1007/BF00042529
H. Le Dret, Incompressible limit behaviour of slightly compressible nonlinear elastic materials, RAIRO Modél. Math. Anal. Numér. 20 (1986), no. 2, 315–340 (English, with French summary). MR 852684, DOI 10.1051/m2an/1986200203151
Patrick Le Tallec, Numerical methods for nonlinear three-dimensional elasticity, Handbook of numerical analysis, Vol. III, Handb. Numer. Anal., III, North-Holland, Amsterdam, 1994, pp. 465–622. MR 1307410, DOI 10.1016/S1570-8659(05)80018-3
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
Frank Natterer, Über die punktweise Konvergenz finiter Elemente, Numer. Math. 25 (1975/76), no. 1, 67–77 (German, with English summary). MR 474884, DOI 10.1007/BF01419529
R. W. Ogden, Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubber-like solids, Proc. Roy. Soc. London, A328 (1972), 567–583.
R. W. Ogden, Volume changes associated with the deformation of rubber-like solids, J. Mech. Phys. Solids 24 (1976), 323–338.
Daniel Pantuso and Klaus-Jürgen Bathe, On the stability of mixed finite elements in large strain analysis of incompressible solids, Finite Elem. Anal. Des. 28 (1997), no. 2, 83–104. MR 1605879, DOI 10.1016/S0168-874X(97)81953-1
M. C. Pélissier, Résolution numérique de quelques problèmes raides en mécanique des milieux faiblement compressibles, Calcolo 12 (1975), no. 3, 275–314 (French, with English summary). MR 421247, DOI 10.1007/BF02576821
S. Reese, P. Wriggers, and B. D. Reddy, A new locking-free brick element technique for large deformation problems in elasticity, Comput. & Structures 75 (2000), no. 3, 291–304. MR 1803813, DOI 10.1016/S0045-7949(99)00137-6
Werner C. Rheinboldt, Methods for solving systems of nonlinear equations, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 14, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. MR 0371057
Rouben Rostamian, Internal constraints in boundary value problems of continuum mechanics, Indiana Univ. Math. J. 27 (1978), no. 4, 637–656. MR 495470, DOI 10.1512/iumj.1978.27.27042
Vitoriano Ruas, Existence and stability of asymmetric finite-element approximations in nonlinear incompressible analysis, SIAM J. Numer. Anal. 26 (1989), no. 5, 1031–1059. MR 1014873, DOI 10.1137/0726058
Alfred H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959–962. MR 373326, DOI 10.1090/S0025-5718-1974-0373326-0
J. C. Simo and M. S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg. 29 (1990), no. 8, 1595–1638. MR 1058742, DOI 10.1002/nme.1620290802
T. Sussman and K. J. Bathe, A finite elment formulation for nonlinear incompressible elastic and noninelastic analysis, Computer & Structures 26 (1987), 357–409.
L. R. Treloar, The Physics of Rubber Elasticity, Oxford University Press, Oxford, 1975.